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Finance Stoch (2016) 20:1097–1108DOI 10.1007/s00780-016-0310-6
No arbitrage of the first kind and local martingalenuméraires
Yuri Kabanov1,2 · Constantinos Kardaras3 ·Shiqi Song4
Received: 3 October 2015 / Accepted: 28 June 2016 / Published online: 21 September 2016© Springer-Verlag Berlin Heidelberg 2016
Abstract A supermartingale deflator (resp. local martingale deflator) multiplica-tively transforms nonnegative wealth processes into supermartingales (resp. localmartingales). A supermartingale numéraire (resp. local martingale numéraire) is awealth process whose reciprocal is a supermartingale deflator (resp. local martin-gale deflator). It has been established in previous works that absence of arbitrage ofthe first kind (NA1) is equivalent to the existence of the (unique) supermartingalenuméraire, and further equivalent to the existence of a strictly positive local martin-gale deflator; however, under NA1, a local martingale numéraire may fail to exist. Inthis work, we establish that under NA1, a supermartingale numéraire under the orig-inal probability P becomes a local martingale numéraire for equivalent probabilitiesarbitrarily close to P in the total variation distance.
Keywords Arbitrage · Viability · Fundamental theorem of asset pricing ·Numéraire · Local martingale deflator · σ -martingale
Mathematics Subject Classification (2010) 91G10 · 60G44
JEL Classification C60 · G13
B Y. [email protected]
1 Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030,Besançon, cedex, France
2 International Laboratory of Quantitative Finance, Higher School of Economics, Moscow, Russia
3 London School of Economics, 10 Houghton Street, London, WC2A 2AE, UK
4 Université d’Evry Val d’Essonne, Boulevard de France, 91037 Evry, cedex, France
1098 Y. Kabanov et al.
1 Introduction
A central structural assumption in the mathematical theory of financial markets is theexistence of so-called local martingale deflators, that is, processes that act multiplica-tively and transform nonnegative wealth processes into local martingales. Under theno free lunch with vanishing risk (NFLVR) condition of [5, 6], the density process ofa local martingale (or, more generally, a σ -martingale) measure is a strictly positivelocal martingale deflator. However, strictly positive local martingale deflators mayexist even if the market allows a free lunch with vanishing risk. Both from a financialand mathematical points of view, an especially important case occurs when a defla-tor is the reciprocal of a wealth process called a local martingale numéraire; in thiscase, the prices of all assets (and in fact, all wealth processes resulting from trading),when denominated in units of the latter local martingale numéraire, are (positive)local martingales.
The relevant, weaker than NFLVR, market viability property, which turns out tobe equivalent to the existence of supermartingale (or local martingale) numéraires,was isolated by various authors under different names: no asymptotic arbitrage ofthe first kind (NAA1), no arbitrage of the first kind (NA1), no unbounded profit withbounded risk (NUPBR), and so on; see [10, 5, 9, 12, 14]. It is not difficult to showthat all these properties are in fact equivalent, even in a wider framework than thatof the standard semimartingale setting—for more information, see the Appendix. Inthe present paper, we opt to utilize the economically meaningful formulation NA1,defined as the property of the market to assign a strictly positive superhedging valueto any non-trivial positive contingent claim.
In the standard financial model studied here, the market is described by a d-dimen-sional semimartingale process S giving the discounted prices of basic securities.In [12], it was shown (even in the more general case of convex portfolio constraints)that the following statements are equivalent:
(i) Condition NA1 holds.(ii) There exists a strictly positive supermartingale deflator.(ii’) The (unique) supermartingale numéraire exists.
In [17], the previous list of equivalent properties was complemented by(iii) There exists a strictly positive local martingale deflator.There are counterexamples (see e.g. [17]) showing that the local martingale nu-
méraire may fail to exist even when there is an equivalent martingale measure (and inparticular when condition NA1 holds). Such examples are possible only in the case ofdiscontinuous asset price processes; it was already shown in [2] that for continuoussemimartingales, among all strictly positive local martingale deflators, there existsone whose reciprocal is a strictly positive wealth process.
In the present note, we add to the above list of equivalences a further property:(iv) In any total-variation neighbourhood of the original probability, there exists
an equivalent probability under which the (unique) local martingale numéraire exists.Establishing the chain (iv) ⇒ (iii) ⇒ (ii) ⇒ (i) is rather straightforward and well
known. The contribution of the note is proving the “closing” implication (i) ⇒ (iv). Itis an obvious corollary of the already known implication (i) ⇒ (ii’) and the followingprincipal result of our note, a version of which was established previously only forthe case d = 1 in [14].
No arbitrage of the first kind 1099
Proposition 1.1 A supermartingale numéraire under P becomes a local martingalenuméraire under probabilities P ∼ P that are arbitrarily close in the total-variationdistance to P .
Proposition 1.1 bears a striking similarity with the density result of σ -martingalemeasures in the set of all separating measures—see [6] and Theorem A.5 in the Ap-pendix. In fact, coupled with certain rather elementary properties of stochastic ex-ponentials, the aforementioned density result is the main ingredient of our proof ofProposition 1.1.
Importantly, we recover in particular the main result of [17], utilising completelydifferent arguments. The proof in [17] combines a change-of-numéraire techniqueand a reduction to the Delbaen–Schachermayer fundamental theorem of asset pricing(FTAP) in [5]. The latter is considered as one of the most difficult and fundamentalresults of arbitrage theory, and the search for simplified proofs still continues—see,for example, [3]. In fact, it may be obtained as a by-product of our result and theversion of the optional decomposition theorem in [16], as has been explained in [13,Sect. 3].
2 Framework and main result
2.1 The setup
In all that follows, we fix T ∈ (0,∞) and work on a filtered probability space(Ω,F ,F = (Ft )t∈[0,T ],P ) satisfying the usual conditions. Unless otherwise explic-itly specified, all relationships between random variables are understood in the P -a.s.sense, and all relationships between stochastic processes are understood moduloP -evanescence.
Let S = (St )t∈[0,T ] be a d-dimensional semimartingale. We denote by L(S) the setof S-integrable processes, that is, the set of all d-dimensional predictable processesfor which the stochastic integral H · S is defined. We stress that we consider generalvector stochastic integration—see [8, Chapter III, Sect. 6].
An integrand H ∈ L(S) such that x + H · S ≥ 0 for some x ∈ R+ will be calledx-admissible. We introduce the set of semimartingales
X x := {H · S : H is x-admissible integrand},
and denote X x> its subset formed by the processes X such that x + X > 0 and
x + X− > 0. These sets are invariant under equivalent changes of the underlyingprobability. Define also the sets of random variables X x
T := {XT : X ∈X x}.For ξ ∈ L0+, we define
x(ξ) := inf{x ∈R+ : there exists X ∈ X x with x + XT ≥ ξ},
with the standard convention inf∅ = ∞.
1100 Y. Kabanov et al.
In the special context of financial modelling:
– The process S represents the price evolution of d liquid assets, discounted by acertain baseline security labelled 0 or d + 1.
– With H being an x-admissible integrand, x + H · S is the value process of a self-financing portfolio with the initial capital x ≥ 0, constrained to stay nonnegative atall times.
– A random variable ξ ∈ L0+ represents a contingent claim, and x(ξ) is its super-hedging value in the class of nonnegative wealth processes.
2.2 Main result
We define |P − P |T V = supA∈F |P [A] − P [A]| as the total-variation distance be-tween the probabilities P and P on (Ω,F).
Theorem 2.1 The following conditions are equivalent:(i) x(ξ) > 0 for every ξ ∈ L0+ \ {0}.(ii) There exists a strictly positive process Y such that the process Y(1 + X) is
a supermartingale for every X ∈ X 1.(iii) There exists a strictly positive process Y with Y0 ∈ L1 such that the process
Y(1 + X) is a local martingale for every X ∈X 1.(iv) For any ε > 0, there exists P ∼ P with |P − P |T V < ε and X ∈ X 1
> such that(1 + X)/(1 + X) is a local P -martingale for every X ∈X 1.
Remark 2.2 It is straightforward to check that statements (ii), (iii), and (iv) of Theo-rem 2.1 are equivalent to the same conditions where “for every X ∈ X 1” is replacedby “for every X ∈X 1
>”.
Theorem 2.1 is formulated in the “pure” language of stochastic analysis. In thecontext of mathematical finance, the following interpretations regarding its statementshould be kept in mind:
– Condition (i) states that any non-trivial contingent claim ξ ≥ 0 has a strictly posi-tive superhedging value. This is referred to as condition NA1 (no arbitrage of thefirst kind); it is equivalent to the boundedness in probability of the set X 1
T or, al-ternatively, to condition NAA1 (no asymptotic arbitrage of the first kind)—see theAppendix.
– The process Y in statement (ii) (resp. in statement (iii)) is called a strictly positivesupermartingale deflator (resp. local martingale deflator).
– A process X with the property in statement (iv) is called a local martingalenuméraire under the probability P . It is well known and not hard to check thata local martingale numéraire (as well as a supermartingale numéraire) is unique ifit exists.
With the above terminology in mind, we may reformulate the properties (i)–(iv)as was done in the Introduction.
No arbitrage of the first kind 1101
3 Proof of Theorem 2.1
3.1 Proof of easy implications
The arguments establishing the implications (iv) ⇒ (iii) ⇒ (ii) ⇒ (i) in Theorem 2.1are elementary and well known; however, for completeness of presentation, we brieflyreproduce them here.
Assume statement (iv), and in its notation fix some ε > 0. Let Z be the densityprocess of P with respect to P , and set Z := 1/(1 + X). For any X ∈ X 1, the processZ(1 + X) is a local P -martingale. Hence, with Y := ZZ, the process Y(1 + X) is alocal P -martingale, that is, (iii) holds.
Since a positive local martingale with initial value in L1 is a supermartingale, theimplication (iii) ⇒ (ii) is obvious.
To establish the implication (ii) ⇒ (i), suppose that Y is a strictly positive super-martingale deflator. It follows that EYT (1 + XT ) ≤ 1 for all X ∈ X 1. Hence, the setYT (1 + X 1
T ) is bounded in L1 and, a fortiori, bounded in probability. Since YT > 0,the set X 1
T is also bounded in probability. The latter property is equivalent to condi-tion NA1—see Lemma A.1 in the Appendix.
By [12, Theorem 4.12] and Lemma A.1 in the Appendix, condition (i) in thestatement of Theorem 2.1 implies the existence of the (unique) supermartingalenuméraire. Therefore, in order to establish the implication (i) ⇒ (iv) of Theorem 2.1and complete its proof, it remains to prove Proposition 1.1. For this, we need someauxiliary facts presented in the next subsection.
3.2 Ratios of stochastic exponentials
We introduce the notation
B(S) := {f ∈ L(S) : f �S > −1};that is, B(S) is the subset of integrands for which the trajectories of the stochasticexponentials E(f · S) are bounded away from zero.
Note that the set 1 +X 1> coincides with the set of stochastic exponentials of inte-
grals with respect to S, that is,
1 +X 1> = {E(f · S) : f ∈ B(S)}.
Indeed, the stochastic exponential corresponding to an integrand f ∈ B(S) is strictlypositive, and so is its left limit, and it satisfies the linear integral equation
E(f · S) = 1 + E−(f · S) · (f · S) = 1 + (E−(f · S)f
) · S.
Thus, E(f ·S) ∈X 1>. Conversely, if the process V = 1 +H ·S is such that V > 0 and
V− > 0, then
V = 1 + (V−V −1− ) · V = 1 + V− · (V −1− · (H · S)) = 1 + V− · ((V −1− H) · S);
that is, V = E(f · S), where f = V −1− H ∈ B(S) because f �S = V −1− �V > −1.
1102 Y. Kabanov et al.
The above observations, coupled with Remark 2.2, imply that condition (iv) maybe alternatively reformulated as follows:
(iv) For any ε > 0, there exist g ∈ B(S) and P ∼ P with |P − P |T V < ε such thatE(f · S)/E(g · S) is a local P -martingale for every f ∈ B(S).
Let Sc denote the continuous local martingale part of the semimartingale S. Recallthat 〈Sc〉 = c · A, where A is a predictable increasing process, and c is a predictableprocess with values in the set of positive semidefinite matrices; then, g ∈ L(Sc) if andonly if |c1/2g|2 · AT < ∞.
In the sequel, fix an arbitrary g ∈ B(S) and set
Sg = S − cg · A −∑
s≤·
gs�Ss
1 + gs�Ss
�Ss. (3.1)
As
∑
s≤·
∣∣∣∣gs�Ss
1 + gs�Ss
∣∣∣∣ |�Ss | ≤ 1
2
∑
s≤T
∣∣∣∣gs�Ss
1 + gs�Ss
∣∣∣∣
2
+ 1
2
∑
s≤T
|�Ss |2 < ∞,
the last term on the right-hand side of (3.1) is a process of finite variation, implyingthat Sg is a semimartingale. (Recall that g ·S is a semimartingale, so that on any finiteinterval, there are P -a.s. only finitely many s with |gs�Ss | ≥ 1/2.)
Noting that �Sg = �S/(1 + g�S), we obtain from (3.1) that
S = Sg + cg · A +∑
s≤·(gs�Ss)�S
gs .
Lemma 3.1 L(S) = L(Sg).
Proof Let f ∈ L(S). Then
|(f, cg)| · AT ≤ 1
2|c1/2f |2 · AT + 1
2|c1/2g|2 · AT < ∞
and
∑
s≤T
|gs�Ssfs�Ss |1 + gs�Ss
≤ 1
2
∑
s≤T
|fs�Ss |2 + 1
2
∑
s≤T
∣∣∣∣
gs�Ss
1 + gs�Ss
∣∣∣∣
2
< ∞.
Thus, L(S) ⊆ L(Sg). To show the opposite inclusion, take f ∈ L(Sg). The conditionsg ∈ L(S) and f ∈ L(Sg) imply that f and g are integrable with respect to Sc = (Sg)c,that is, |c1/2g|2 · AT < ∞ and |c1/2f |2 · AT < ∞. As previously, it then follows that|(f, cg)| · AT < ∞. Since also
∑
s≤t
|(gs�Ss)(fs�Sgs )| ≤ 1
2
∑
s≤T
|gs�Ss |2 + 1
2
∑
s≤T
|fs�Sgs |2 < ∞,
we obtain that f ∈ L(S), that is, the inclusion L(Sg) ⊆ L(S). �
Lemma 3.2 B(Sg) = B(S) − g.
No arbitrage of the first kind 1103
Proof Let h = f − g, where f ∈ B(S). Then, h ∈ L(S) = L(Sg) by Lemma 3.1, and
h�Sg = (f − g)�Sg = (f − g)�S
1 + g�S= 1 + f �S
1 + g�S− 1 > −1.
Conversely, start with h ∈ B(Sg). Then, using again Lemma 3.1, we obtain thatf := h+g belongs to L(S). Moreover, recalling the relation �S = �Sg/(1 − g�Sg),we obtain that
f �S = (h + g)�S = (h + g)�Sg
1 − g�Sg= 1 + h�Sg
1 − g�Sg− 1 > −1,
which completes the proof. �
For f ∈ B(S), Lemma 3.2 gives f −g ∈ B(Sg); then, straightforward calculationsusing Yor’s product formula
E(U)E(V ) = E(U + V + [U,V ]),which is valid for arbitrary semimartingales U and V , lead to the identity
E(f · S)
E(g · S)= E
((f − g) · Sg
).
(In this regard, see also [12, Lemma 3.4].) Then, invoking Lemma 3.2, we obtain theset equality
1 +X 1>(Sg) = E−1(g · S)
(1 +X 1
>(S)).
3.3 Proof of Proposition 1.1
Let the process E(g ·S), where g ∈ B(S), be the (unique) supermartingale numéraire;in other words, the ratio E(f · S)/E(g · S) is a supermartingale for each f ∈ B(S).Passing to Sg and using Lemma 3.2, we obtain that EET (h ·Sg) ≤ 1 for all h ∈ B(Sg).Therefore, EH ·Sg
T ≤ 0 for every H ∈ L(Sg) such that H ·Sg > −1 and H ·Sg− > −1
and thus for every H ∈ L(Sg) for which the process H · Sg is bounded from below.This means, in the terminology of [9], that the probability P is a separating measurefor Sg . An application of [6, Proposition 4.7] (or Theorem A.5) implies, for anyε > 0, the existence of probability P ∼ P , depending on ε, with |P − P |T V < ε andsuch that Sg is a σ -martingale with respect to P , that is, Sg = G · M where G isa (0,1]-valued one-dimensional predictable process and M is a d-dimensional localP -martingale. Recall that a bounded from below stochastic integral with respect to alocal martingale is a local martingale [1, Proposition 3.3]. The ratio E(f ·S)/E(g ·S),being an integral with respect to Sg , hence with respect to M , is a P -local martingalefor each f ∈ B(S), which is exactly what we need. �
Remark 3.1 An inspection of the arguments in [12] used to establish the implica-tion (i) ⇒ (ii’) reveals that in the case where the (random, (w, t)-dependent) Lévymeasures of S are concentrated on finite sets also depending on (ω, t), the (unique)supermartingale numéraire is in fact the (unique) local martingale numéraire.
1104 Y. Kabanov et al.
Acknowledgements The research of Yuri Kabanov is funded by the grant of the Government of RussianFederation No. 14.12.31.0007. The research of Constantinos Kardaras is partially funded by the MC grantFP7-PEOPLE-2012-CIG, 334540. The authors would like to thank three anonymous referees for theirhelpful remarks and suggestions.
Appendix: No-arbitrage conditions, revisited
A.1 Condition NA1: equivalent formulations
We discuss equivalent forms of the condition NA1 in the context of a general abstractsetting, where the model is given by specifying the set of wealth processes. The ad-vantage of this generalization is that we can use only elementary properties withoutany reference to stochastic calculus and integration theory.
Let X 1 be a convex set of càdlàg processes X with X ≥ −1 and X0 = 0, contain-ing the zero process. For x ≥ 0, we define the set X x = xX 1 and note that X x ⊆ X 1
when x ∈ [0,1]. Put X := coneX 1 = R+X 1 and define the sets of terminal randomvariables X 1
T := {XT : X ∈ X 1} and XT := {XT : X ∈ X }. In this setting, the ele-ments of X are interpreted as admissible wealth processes starting from zero initialcapital; the elements of X x are called x-admissible.
Remark A.1 (“Standard” model) In the typical example, a d-dimensional semimartin-gale S is given and X 1 is the set of stochastic integrals H ·S, where H is S-integrableand H · S ≥ −1. Though our main result deals with the standard model, it is naturalto discuss the basic definitions and their relations with concepts of arbitrage theoryin a more general framework.
Define the set of strictly 1-admissible processes X 1> ⊆ X 1 as composed of those
X ∈ X 1 such that X > −1 and X− > −1. The sets x + X x , x + X x>, and so on,
x ∈ R+, have obvious interpretations. We are particularly interested in the set 1+X 1>.
Its elements are strictly positive wealth processes starting with unit initial capital andmay be thought of as tradable numéraires.
For ξ ∈ L0+, define the superhedging value x(ξ) := inf{x : ξ ∈ x +X xT −L0+}. We
say that the wealth-process family X satisfies the condition NA1 (no arbitrage of thefirst kind) if x(ξ) > 0 for every ξ ∈ L0+ \ {0}. Alternatively, the condition NA1 can bedefined via
( ⋂
x>0
{x +X x
T − L0+}) ∩ L0+ = {0}.
The family X is said to satisfy the condition NAA1 (no asymptotic arbitrage ofthe first kind) if for any sequence (xn)n of positive numbers with xn ↓ 0 and anysequence of value processes Xn ∈X such that xn + Xn ≥ 0, we have
lim supn→∞
P [xn + XnT ≥ 1] = 0.
Finally, we say that the family X satisfies the condition NUPBR (no unboundedprofit with bounded risk) if the set {XT : X ∈ X 1
>} is bounded in L0. Since we have
No arbitrage of the first kind 1105
(1/2)X 1T = X 1/2
T ⊆ {XT : X ∈ X 1>}, the set {XT : X ∈ X 1
>} is bounded in L0 if andonly if the set X 1
T is bounded in L0.The next result shows that all three market viability notions introduced above co-
incide.
Lemma A.1 NAA1 ⇐⇒ NUPBR ⇐⇒ NA1.
Proof NAA1 ⇒ NUPBR: If the family {XT : X ∈ X 1>} fails to be bounded in L0,
then P [1 + XnT ≥ n] ≥ ε > 0 for a sequence (Xn) ⊆ X 1
>, and we obtain a violation ofNAA1 with n−1 + n−1Xn
T .NUPBR ⇒ NA1: If NA1 fails, then there exist ξ ∈ L0+ \ {0} and a sequence (Xn)
with Xn ∈ X 1/n such that 1/n + Xn ≥ ξ . Then the sequence nXnT ∈ X 1 fails to be
bounded in L0, in violation of NUPBR.NA1 ⇒ NAA1: If the implication fails, then there are sequences xn ↓ 0 and
Xn ≥ −xn such that P [xn + XnT ≥ 1] ≥ 2ε > 0. By the von Weizsäcker theorem (see
[18] or [11, Theorem A.2.3]), any sequence of random variables bounded from belowcontains a subsequence converging in the Cesarò sense a.s. as well as all its furthersubsequences. We may assume without loss of generality that for ξn := xn +Xn
T , thesequence ξ n := (1/n)
∑ni=1 ξi converges to ξ ∈ L0+. Note that ξ �= 0. Indeed,
ε(1 − P [ξ n ≥ ε]) ≥ 1
n
n∑
i=1
EξiI{ξ n<ε} ≥ 1
n
n∑
i=1
EξiI{ξ i≥1, ξ n<ε}
≥ 1
n
n∑
i=1
P [ξ i ≥ 1, ξ n < ε] ≥ 1
n
n∑
i=1
(P [ξ i ≥ 1] − P [ξ n ≥ ε])
≥ 2ε − P [ξ n ≥ ε].It follows that P [ξ n ≥ ε] ≥ ε/(1 − ε). Thus,
E[ξ ∧ 1] = limn
E[ξ n ∧ 1] ≥ ε2/(1 − ε) > 0.
It follows that there exists a > 0 such that P [ξ ≥ 2a] > 0. In view of Egorov’s theo-rem, we can find a measurable set Γ ⊆ {ξ ≥ a} with P [Γ ] > 0 on which xn +Xn ≥ a
for all sufficiently large n. But this means that the random variable aIΓ �= 0 can besuperreplicated starting with arbitrarily small initial capital, in contradiction with theassumed condition NA1. �
Remark A.2 (On terminology and bibliography) The conditions NAA1 and NA1 haveclear financial meanings, whereas the boundedness in L0 of the set X 1
T , at first glance,looks like a technical condition—see [5]. The concept of NAA1 first appeared in [10]in a much more general context of large financial markets, along with another funda-mental notion NAA2 (no asymptotic arbitrage of the second kind). The boundednessin L0 of X 1
T was discussed in [9] (as the BK-property), in the framework of a modelgiven by value processes; however, it was overlooked that it coincides with NAA1 forthe “stationary” model, that is, when the stochastic basis and the price process do not
1106 Y. Kabanov et al.
depend on n. The same condition appeared under the acronym NUPBR in [12] andwas shown to be equivalent to NA1 in [13].
A.2 NA1 and NFLVR
Remaining in the framework of the abstract model of the previous subsection, weprovide here results on the relation of the condition NA1 with other fundamentalnotions of arbitrage theory; compare with [9].
Define the convex sets C := (XT − L0+) ∩ L∞ and denote by C and C∗ the norm-closure and weak∗ closure of C in L∞, respectively. The conditions NA, NFLVR,and NFL are defined correspondingly via
C ∩ L∞+ = {0}, C ∩ L∞+ = {0}, C∗ ∩ L∞+ = {0}.Consecutive inclusions induce a hierarchy of these properties via
C ⊆ C ⊆ C∗,NA ⇐= NFLVR ⇐= NFL.
Lemma A.2 NFLVR =⇒ NA&NA1.
Proof Assume that NFLVR holds. Condition NA follows trivially. If NA1 fails, thenthere exists a [0,1]-valued ξ ∈ L0+ \ {0} such that for each n ≥ 1, we can findXn ∈ X 1/n with 1/n + Xn
T ≥ ξ . Then the random variables XnT ∧ ξ belong to C
and converge uniformly to ξ , contradicting NFLVR. �
To obtain the converse implication in Lemma A.2, we need an extra property. Wecall a model natural if the elements of X are adapted processes and for any X ∈ X ,s ∈ [0, T ) and Γ ∈Fs , the process X := IΓ ∩{Xs≤0}I[s,T ](X−Xs) is an element of X .In words, a model is natural if an investor deciding to start trading at time s when theevent Γ happened can use from this time, if Xs ≤ 0, an investment strategy that leadsto a value process with the same increments as X.
Lemma A.3 Suppose that the model is natural. If NA holds, then any X ∈ X admitsthe bound X ≥ −λ, where λ = ‖X−
T ‖∞.
Proof If P [Xs < −λ] > 0, then X := I{Xs<−λ}I[s,T ](X − Xs) belongs to X and sat-isfies XT ≥ 0 and P [XT > 0] > 0, in violation of NA. �
Proposition A.4 Suppose that the model is natural and in addition that for everyn ≥ 1 and X ∈X with X ≥ −1/n, the process nX is in X 1. Then
NFLVR ⇐⇒ NA & NA1.
Proof By Lemma A.2, we only have to show the implication “⇐”. If NFLVR fails,then there are ξn ∈ C and ξ ∈ L∞+ \ {0} such that ‖ξn − ξ‖∞ ≤ n−1. By definition, wehave ξn ≤ ηn = Xn
T , where Xn ∈ X . Obviously, ‖η−n ‖∞ ≤ n−1, and since NA holds,
No arbitrage of the first kind 1107
nXn ∈ X 1 by virtue of Lemma A.3 and our hypothesis. By the von Weizsäcker theo-rem, we may assume that ηn → η a.s. Since P [η > 0] > 0, the sequence nXn
T ∈ X 1T
tends to infinity with strictly positive probability, violating condition NUPBR or,equivalently, NA1. �
Examples showing that the conditions NFLVR, NA, and NA1 are all different,even for the standard model (satisfying, of course, the hypotheses of the above propo-sition) can be found in [7].
Assume now that X 1 is a subset of the space S of semimartingales, equipped withthe Emery topology given by the quasi-norm
D(X) := sup{E[1 ∧ |H · XT |] : H is predictable, |H | ≤ 1}.Define the condition ESM as the existence of a probability P ∼ P such that EXT ≤ 0for all processes X ∈ X . A probability P with this property is referred to as an equiv-alent separating measure. According to the Kreps–Yan separation theorem [11, The-orem 2.1.4], the conditions NFL and ESM are equivalent. The next result is provedin [9] on the basis of the paper [5], where this theorem was established for the “stan-dard” model; see also [3].
Theorem A.3 Suppose that X 1 is closed in S and that the following concatenationproperty holds: for any X,X′ ∈ X 1 and any bounded predictable processes H,G ≥ 0such that HG = 0, the process X := H · X + G · X′ belongs to X 1 if it satisfies theinequality X ≥ −1. Then, under the condition NFLVR, it holds that C = C∗, and asa corollary, we have
NFLVR ⇐⇒ NFL ⇐⇒ ESM.
Remark A.4 It is shown in [15, Theorem 1.7] that the condition NA1 is equivalentto the existence of the (unique) supermartingale numéraire in a setting where thewealth-process sets are abstractly defined via a requirement of predictable convexity(also called fork-convexity).
In the case of the “standard” model with a finite-dimensional semimartingale S de-scribing the prices of the basic risky securities, we have the following: If S is bounded(resp. locally bounded), a separating measure is a martingale measure (resp. localmartingale measure). Without any local boundedness assumption on S, we have thefollowing result from [6], a short proof of which is given in [9] and which we usehere.
Theorem A.5 In any neighbourhood in total variation of a separating measure, thereexists an equivalent probability under which S is a σ -martingale.
It follows that if NFLVR holds, then the process S is a σ -martingale with respectto some probability measure P ′ ∼ P with density process Z′. Therefore, for anyprocess X = H · S from X 1, the process 1 + X is a local martingale with respectto P ′, or equivalently, Z′(1 + X) is a local martingale with respect to P ; therefore,Z′ is a local martingale deflator.
1108 Y. Kabanov et al.
Remark A.6 A counterexample in [4, Sect. 6] involving a simple one-step modelshows that Theorem A.5 is not valid in markets with countably many assets. As acorollary, the condition NFLVR (equivalent in this one-step model to NA1) is notsufficient to ensure the existence of a local martingale measure or a local martingaledeflator.
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